Katrin FAESSLER
Katrin FAESSLER
Katrin FAESSLER
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Analytic characterization of QC maps on H 1<br />
Theorem (Korányi, Reimann, Pansu, Vodop’yanov)<br />
A homeomorphism f : Ω → Ω ′ between domains in H 1 is quasiconformal if<br />
and only if<br />
f ∈ HW 1,4<br />
loc (Ω),<br />
f is a weakly contact map,<br />
Here,<br />
∃ 1 ≤ K < ∞ such that DHf (p) 4 ≤ KJ(p, f ) a.e.<br />
DHf 4 = (|ZfI | + |¯ZfI |) 4<br />
and J(·, f ) = (|ZfI | 2 − |¯ZfI | 2 ) 2 .<br />
Definition (Distortion)<br />
<br />
<br />
|ZfI<br />
K(·, f ) = <br />
| + |ZfI | <br />
<br />
<br />
|ZfI | − |ZfI |<br />
and Kf = K(·, f )∞.<br />
Note: H(·, f ) = K(·, f ) a.e.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 13 / 23
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